Non-classical solutions of the $p$-Laplace equation

In this paper we settle Iwaniec and Sbordone’s 1994 conjecture concerning very weak solutions to the p -Laplace equation. Namely, on the one hand we show that distributional solutions of the p -Laplace equation in W^{1,r} for p \neq 2 and r>\max\,\{ 1,p-1\} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone’s conjecture in general.